On pointwise second‐order maximum principle for optimal stochastic controls of general mean‐field type

Abstract

AbstractIn this paper, we establish a second‐order stochastic maximum principle for optimal stochastic control of stochastic differential equations of general mean‐field type. The coefficients of the system are nonlinear and depend on the state process as well as of its probability law. The control variable is allowed to enter into both drift and diffusion terms. We establish a set of second‐order necessary conditions for the optimal control in integral form. The control domain is assumed to be convex. The proof of our main result is based on the first‐ and second‐order derivatives with respect to the probability law and by using a convex perturbation with some appropriate estimates.